How do you tell the moving direction of a wave?

[KFC]

For a 1D plane wave written mathematically as

$$\exp(i k x)$$

where i is sqrt(-1). k is the wavenumber. In many textbook, it reads this wave is moving to the right while $$\exp(-i k x)$$ is moving to the left. It is quite confusing. How do you tell that?

 

[jtbell]

Those expressions specify only the spatial part of the wave function. They don't contain any time dependence so you can't say anything about the direction of propagation without making further assumptions.

 

[KFC]

Those expressions specify only the spatial part of the wave function. They don't contain any time dependence so you can't say anything about the direction of propagation without making further assumptions.

Thanks for reply. But why in the textbook, in the chapter about scattering, it keeps saying $$e^{ikz}$$ is the incoming wave propagate along positive z direction.

 

[jtbell]

Which course is this, by the way? You mention scattering, so is this a quantum mechanics course?

In that case, they are probably assuming that the time dependence is $e^{- i \omega t} = e^{- i E t / \hbar}$. In that case the complete wave functions are

$$e^{i(kx - \omega t)} = \exp \left[ ik \left( x - \frac{\omega}{k} t \right) \right] = \exp \left[ ik \left( x - v t \right) \right]$$

$$e^{i(-kx - \omega t)} = \exp \left[ -ik \left( x + \frac{\omega}{k} t \right) \right] = \exp \left[ -ik \left( x + v t \right) \right]$$

In general, a function of the form f(x-vt) represents a wave traveling in the +x direction, and f(x+vt) represents a wave traveling in the -x direction. These are the most general solutions to the differential wave equation in one dimension.